AP Precalculus Practice Test: Unit 3 Question #27 Creating a Sinusoidal Function

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**AP Precalculus Practice Test: Unit 3, Question #27** asks you to create a sinusoidal function from given details about its behavior, such as maximum and minimum values, period, and starting position. The function is written in the form \( y = a \sin(b(x - h)) + k \) or \( y = a \cos(b(x - h)) + k \), where each parameter represents specific characteristics of the function.

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### **Key Steps for Creating a Sinusoidal Function**
1. **Amplitude (\(a\))**:
- The amplitude is the distance from the midline to the maximum or minimum value.
- It is calculated as half the difference between the maximum and minimum values.

2. **Vertical Shift (\(k\))**:
- The vertical shift is the location of the midline of the function, calculated as the midpoint between the maximum and minimum values.

3. **Period and Frequency (\(b\))**:
- The period is the horizontal length of one full cycle of the function.
- Use the formula \( b = \frac{2\pi}{\text{Period}} \) to calculate \(b\).

4. **Horizontal Shift (\(h\))**:
- The horizontal shift adjusts where the cycle starts.
- This depends on whether the function starts at the maximum, minimum, or midline and whether it is increasing or decreasing.

5. **Choose Sine or Cosine**:
- Use the sine function if the graph starts at the midline.
- Use the cosine function if the graph starts at the maximum or minimum.

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### **Example Problem**
**Given Information**:
- Maximum value is 8.
- Minimum value is 4.
- The period is 6.
- The graph starts at the maximum.

1. **Amplitude**:
- The difference between the maximum and minimum is 4. Divide by 2 to get an amplitude of 2.

2. **Vertical Shift**:
- The midpoint between 8 and 4 is 6, so the vertical shift is 6.

3. **Period and \(b\)**:
- The period is 6. Using \( b = \frac{2\pi}{\text{Period}} \), we calculate \( b = \frac{2\pi}{6} = \frac{\pi}{3} \).

4. **Horizontal Shift**:
- The graph starts at the maximum, so there is no horizontal shift (\( h = 0 \)).

5. **Function**:
- Since the graph starts at the maximum, use the cosine function. The equation is:
\[
y = 2 \cos\left(\frac{\pi}{3}x\right) + 6
\]

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### **Description**:
"The sinusoidal function has an amplitude of 2, a vertical shift of 6, and a period of 6. The graph starts at its maximum value, so a cosine function is used. The resulting equation is \( y = 2 \cos\left(\frac{\pi}{3}x\right) + 6 \)."

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Nick Perich
Norristown Area High School
Norristown Area School District
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